A Superconvergent Ensemble HDG Method for Parameterized Convection Diffusion Equations

TL;DR

This paper introduces an ensemble HDG method for efficiently solving multiple parameterized convection-diffusion PDEs, achieving optimal convergence and superconvergence rates with confirmed numerical experiments.

Contribution

The paper develops a novel ensemble HDG method with proven optimal and superconvergent rates for parameterized PDEs, including the first such $L^2$ convergence result in the literature.

Findings

Optimal $L^2$ convergence rate established

Superconvergence achieved after postprocessing

Numerical experiments confirm theoretical results

Abstract

In this paper, we first devise an ensemble hybridizable discontinuous Galerkin (HDG) method to efficiently simulate a group of parameterized convection diffusion PDEs. These PDEs have different coefficients, initial conditions, source terms and boundary conditions. The ensemble HDG discrete system shares a common coefficient matrix with multiple right hand side (RHS) vectors; it reduces both computational cost and storage. We have two contributions in this paper. First, we derive an optimal $L^2$ convergence rate for the ensemble solutions on a general polygonal domain, which is the first such result in the literature. Second, we obtain a superconvergent rate for the ensemble solutions after an element-by-element postprocessing under some assumptions on the domain and the coefficients of the PDEs. We present numerical experiments to confirm our theoretical results.